SetDirectory["~/KappaLib"];
<<kappaLib-1.1.m

Metaclass I

kappa=emMatrixToKappa[(({{a1, 0, 0, -b1, 0, 0}, {0, a2, 0, 0, -b2, 0}, {0, 0, a3, 0, 0, -b3}, {b1, 0, 0, a1, 0, 0}, {0, b2, 0, 0, a2, 0}, {0, 0, b3, 0, 0, a3}}))];

We can exclude the case a1=a2=a3, b1=b2=b3

fr=emKappaToFresnel[kappa,{x0,x1,x2,x3}]//.{a3->a1,a2->a1,b3->b1,b2->b1};
FullSimplify[fr]

Define D1,D2,D3 as functions of kappa

D1[kappa_]:=Module[
{m,a1,a2,a3,b1,b2,b3},
m=emKappaToMatrix[kappa];
a1=m[[1]][[1]];a2=m[[2]][[2]];a3=m[[3]][[3]];
b1=m[[1]][[4]];b2=m[[2]][[5]];b3=m[[3]][[6]];
Return[((((a2-a3))^2)+(b2^2)+(b3^2))/(b2 b3)];
];
D2[kappa_]:=Module[
{m,a1,a2,a3,b1,b2,b3},
m=emKappaToMatrix[kappa];
a1=m[[1]][[1]];a2=m[[2]][[2]];a3=m[[3]][[3]];
b1=m[[1]][[4]];b2=m[[2]][[5]];b3=m[[3]][[6]];
Return[((((a1-a3))^2)+(b1^2)+(b3^2))/(b1 b3)];

];
D3[kappa_]:=Module[
{m,a1,a2,a3,b1,b2,b3},
m=emKappaToMatrix[kappa];
a1=m[[1]][[1]];a2=m[[2]][[2]];a3=m[[3]][[3]];
b1=m[[1]][[4]];b2=m[[2]][[5]];b3=m[[3]][[6]];
Return[((((a1-a2))^2)+(b1^2)+(b2^2))/(b1 b2)];
];

D1[kappa]
D2[kappa]
D3[kappa]

We can exchange D1 <-> D2 by an orientation preserving coordinate change

J=(({{1, 0, 0, 0}, {0, 0, -1, 0}, {0, 1, 0, 0}, {0, 0, 0, 1}}));
Det[J]
kappaP=emCoordinateChange[kappa,J];

emKappaToMatrix[kappaP]//MatrixForm

Simplify[D2[kappa]-D1[kappaP]]
Simplify[D1[kappa]-D2[kappaP]]
Simplify[D3[kappa]-D3[kappaP]]

We can exchange D2 <-> D3 by an orientation preserving coordinate change

J=(({{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, -1}, {0, 0, 1, 0}}));
Det[J]
kappaP=emCoordinateChange[kappa,J];

emKappaToMatrix[kappaP]//MatrixForm

Simplify[D1[kappa]-D1[kappaP]]
Simplify[D2[kappa]-D3[kappaP]]
Simplify[D3[kappa]-D2[kappaP]]